computing marginal distributions, Farlie-Gumbel-Morgenstern

Question

Let $h: \mathbb R^2 \rightarrow \mathbb R, h(u,v) = (1 + a(2u-1)(2v-1)) I_{[0,1]}(u) I_{[0,1]}(v)$ the probability density function of the Farlie-Gumbel-Morgenstern-Copula, $|a| \leq 1$, $F_X, F_Y$ diffeomorphisms $ \mathbb R \rightarrow (0,1)$.

I know that $f: \mathbb R \times \mathbb R, f(x,y) = f_X(x)f_Y(y)h(F_X(x),F_Y(y))$ is the probability density function of $(X,Y)=(F^{-1}_X(U),F^{-1}_Y(V))$, if $(U,V)$ has probability density function $h$.

Now I want to compute the marginal distributions, so the distributions of $X$ and $Y$, so I have to integrate $f(x,y)dy$ and $f(x,y)dx$, but I don't know how to do that. Can anybody help? And what is an example of a probability density function which has the same marginal distributions as $f$, but defines another distribution?

Thanks in advance!

Answer 1

\begin{align} \int_{\mathbb{R}} f(x,y)dx=&\int_{\mathbb{R}} f_{X}(x)f_{Y}(y)dx+a(2f_{Y}(y)F_{Y}(y)-f_{Y}(y))\int_{\mathbb{R}} (2f_{X}(x)F_{X}(x)-f_{X}(x))dx\\ =&f_{Y}(y)\int_{\mathbb{R}} f_{X}(x)dx+a(2f_{Y}(y)F_{Y}(y)-f_{Y}(y))\big(\big[F_{X}^{2}(x)\big]_{-\infty}^{\infty}-\int_{\mathbb{R}} f_{X}(x)dx\big)\\ =&f_{Y}(y)+a(2f_{Y}(y)F_{Y}(y)-f_{Y}(y))(1-1)\\ =&f_{Y}(y). \end{align} Similarly, $\int_{\mathbb{R}} f(x,y)dy=f_{X}(x).$